The Median of a Set of Data

The Median of a Set of Data

Median exampleProblem: The Doran family has 5 children, aged 9, 12, 7, 16 and 13. What is the age of the middle child?

Solution: Ordering the childrens' ages from least to greatest, we get:

7,  9,  12,  13,  16

Answer: The age of the middle child is the middlemost number in the data set, which is 12.

In the problem above, we found the median of a set of 5 numbers.

Definition: The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should first be arranged in order from least to greatest.

To remember the definition of a median, just think of the median of a road, which is the middlemost part of the road. In the problem above, 12 is the median: it is the number that is halfway into the set. There are two children older than 12 and two children younger than 12. Let's look at some more examples.


Gas pumpExample 1: The Jameson family drove through 7 states on their summer vacation. Gasoline prices varied from state to state. What is the median gasoline price?

$1.79,  $1.61,  $1.96,  $2.09,  $1.84,  $1,75,  $2.11

Solution: Ordering the data from least to greatest, we get:

$1.61,  $1.75,  $1.79,  $1.84,  $1.96,  $2.09,  $2.11

Answer: The median gasoline price is $1.84. (There were 3 states with higher gasoline prices and 3 with lower prices.)


Test paperExample 2: During the first marking period, Nicole's math quiz scores were 90, 92, 93, 88, 95, 88, 97, 87, and 98. What was the median quiz score?

Solution: Ordering the data from least to greatest, we get:

87,  88,  88,  90,  92,  93,  95,  96,  98

Answer: The median quiz score was 92. (Four quiz scores were higher than 92 and four were lower.)


In each of the examples above, there is an odd number of items in each data set. In Example 1, there are 7 numbers in the data set; in Example 2 there are 9 numbers. Let's look at some examples in which there is an even number of items in the data set.

MarathonExample 3: A marathon race was completed by 4 participants. What was the median race time?

2.7 hr,  8.3 hr,  3.5 hr,  5.1 hr

Solution: Ordering the data from least to greatest, we get:

2.7,  3.5,  5.1,  8.3

Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers.

3.5 + 5.1 = 8.6

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Answer: The median race time was 4.3 hr.


BonusExample 4: The salaries of 8 employees who work for a small company are listed below. What is the median salary?

$40,000, $29,000, $35,500, $31,000, $43,000, $30,000, $27,000, $32,000

Solution: Ordering the data from least to greatest, we get:

$27,000,  $29,000,  $30,000,  $31,000,  $32,000,  $35,500,  $40,000,  $43,000

Since there is an even number of items in the data set, we compute the median by taking the mean of the two middlemost numbers.

$31,000 + $32,000 = $63,000

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Answer: The median salary is $31,500.


Summary: The median of a set of data is the middlemost number in the set. The median is also the number that is halfway into the set. To find the median, the data should be arranged in order from least to greatest. If there is an even number of items in the data set, then the median is found by taking the mean (average) of the two middlemost numbers.


Exercises

Directions: Find the median of each set of data. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

1. Find the median weight of 5 seventh grade students whose weights in pounds are listed below.
82 lb,  102 lb,  75 lb,  91 lb,  89 lb
ANSWER BOX:  lb  

RESULTS BOX: 

2. The ages of 10 college students are listed below. Find the median.
18,  24,  20,  35,  19,  23,  26,  23,  19,  20
ANSWER BOX:   

RESULTS BOX: 

3. Seven teenagers receive the allowances listed below. What is the median allowance?
$15,  $10,  $12,  $18,  $14,  $15,  $8
ANSWER BOX:  $   

RESULTS BOX: 

4. Find the median number of magazines purchased in a store by 6 customers.
1,  7,  4,  2,  3,  4. 
ANSWER BOX:   

RESULTS BOX: 

5. The annual snowfall in inches for New York City for the last 9 years is listed below. Find the median snowfall.
20.5 in,  26.1 in,  32.5 in,  18.9 in,  33.4 in,  29.7 in,  19.8 in,  25.6 in,  34.3 in 
ANSWER BOX:  in  

RESULTS BOX: